At first login for each user, KDE copies files from /usr/share/apps/kdesktop/DesktopLinks to the Desktop directory of the user. This means that if there is a Home Icon in DesktopLinks and you add a Home Icon via TLDC, there will be two Home Icons.
Remove the contents of /usr/share/apps/kdesktop/DesktopLinks to solve the problem, and let TLDC be the sole provider of icons on the desktop.
If you KDE is based somewhere else than under /usr, the DesktopLinks directory will be situated elesewhere. For example, on SuSE, KDE is based at /opt/kde3.
When KDE tries to determine what application to use for opening a specific file, it is only looking for applications that are available in the menu. There are cases where not all applications that may be used for opening files are meant to be available in the menu.
In this case, create a hidden menu by setting "Hide this Menu" to Yes in the Menu Structure Editor, and then create an Application Group that adds the applications that should be available for file associations in to this menu.
This is a case of the problem above where File Associations are not working. Create an Application Group that includes the Konqueror (kde-kfmclient_dir) application in a hidden menu, and make sure this application group is added for all relevant users, and the home icon will work again.
A menu named "Hidden Menu" is created by the application group "Hidden" which is by default activated for the profile kde. This menu contains the kde-kcmclient_dir) to make sure the home icon is working. Make sure this application group is activated for all users with a desktop based on KDE.